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\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  -5.0 & -1.00 x_{1} & -7.00 x_{2} & +  8.00 x_{3} & -1.00 x_{4} & +  9.00 x_{5}\\
 x_{7}   &  22.0 & -5.00 x_{1} &   & -8.00 x_{3} & +  3.00 x_{4} & -8.00 x_{5}\\
 x_{8}   &  20.0 & -9.00 x_{1} & -10.00 x_{2} & -3.00 x_{3} & -4.00 x_{4} & -3.00 x_{5}\\
 x_{9}   &  0.0 & +  2.00 x_{1} & +  1.00 x_{2} & -3.00 x_{3} & +  7.00 x_{4} & -2.00 x_{5}\\
 x_{10}   &  14.0 & -8.00 x_{1} & -7.00 x_{2} & -1.00 x_{3} & -10.00 x_{4} & +  5.00 x_{5}\\
\hline
z    &  0.0  &   & -1.00 x_{2} & -1.00 x_{3} &    &   \\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{5}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,10$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & +  1.00 y_{6} & +  5.00 y_{7} & +  9.00 y_{8} & -2.00 y_{9} & +  8.00 y_{10}\\
 y_{2}   &  1.0 & +  7.00 y_{6} &   & + 10.00 y_{8} & -1.00 y_{9} & +  7.00 y_{10}\\
 y_{3}   &  1.0 & -8.00 y_{6} & +  8.00 y_{7} & +  3.00 y_{8} & +  3.00 y_{9} & +  1.00 y_{10}\\
 y_{4}   &  1.0 & +  1.00 y_{6} & -3.00 y_{7} & +  4.00 y_{8} & -7.00 y_{9} & + 10.00 y_{10}\\
 y_{5}   &  1.0 & -9.00 y_{6} & +  8.00 y_{7} & +  3.00 y_{8} & +  2.00 y_{9} & -5.00 y_{10}\\
\hline
z    &  -0 & +  5.00 y_{6} & -22.00 y_{7} & -20.00 y_{8} &   & -14.00 y_{10}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 3 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  0.819672131148 & -0.08 y_{5} & +  6.44 y_{7} & +  8.20 y_{8} & +  0.26 y_{4} & +  4.97 y_{10}\\
 y_{2}   &  1.86885245902 & -0.79 y_{5} & +  6.05 y_{7} & + 12.69 y_{8} & -0.08 y_{4} & +  3.89 y_{10}\\
 y_{3}   &  0.311475409836 & +  0.87 y_{5} & +  0.51 y_{7} & +  1.11 y_{8} & -0.18 y_{4} & +  7.15 y_{10}\\
 y_{9}   &  0.16393442623 & -0.02 y_{5} & -0.31 y_{7} & +  0.64 y_{8} & -0.15 y_{4} & +  1.39 y_{10}\\
 y_{6}   &  0.147540983607 & -0.11 y_{5} & +  0.82 y_{7} & +  0.48 y_{8} & -0.03 y_{4} & -0.25 y_{10}\\
\hline
z    &  0.737704918033 & -0.57 y_{5} & -17.90 y_{7} & -17.62 y_{8} & -0.16 y_{4} & -15.23 y_{10}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  0.573770491803 & +  0.08 x_{1} & +  0.79 x_{2} & -0.87 x_{3} & +  0.02 x_{9} & +  0.11 x_{6}\\
 x_{7}   &  17.9016393443 & -6.44 x_{1} & -6.05 x_{2} & -0.51 x_{3} & +  0.31 x_{9} & -0.82 x_{6}\\
 x_{8}   &  17.6229508197 & -8.20 x_{1} & -12.69 x_{2} & -1.11 x_{3} & -0.64 x_{9} & -0.48 x_{6}\\
 x_{4}   &  0.16393442623 & -0.26 x_{1} & +  0.08 x_{2} & +  0.18 x_{3} & +  0.15 x_{9} & +  0.03 x_{6}\\
 x_{10}   &  15.2295081967 & -4.97 x_{1} & -3.89 x_{2} & -7.15 x_{3} & -1.39 x_{9} & +  0.25 x_{6}\\
\hline
z    &  -0.737704918033 & -0.82 x_{1} & -1.87 x_{2} & -0.31 x_{3} & -0.16 x_{9} & -0.15 x_{6}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  0.573770491803 & +  0.08 x_{1} & +  0.79 x_{2} & -0.87 x_{3} & +  0.02 x_{9} & +  0.11 x_{6}\\
 x_{7}   &  17.9016393443 & -6.44 x_{1} & -6.05 x_{2} & -0.51 x_{3} & +  0.31 x_{9} & -0.82 x_{6}\\
 x_{8}   &  17.6229508197 & -8.20 x_{1} & -12.69 x_{2} & -1.11 x_{3} & -0.64 x_{9} & -0.48 x_{6}\\
 x_{4}   &  0.16393442623 & -0.26 x_{1} & +  0.08 x_{2} & +  0.18 x_{3} & +  0.15 x_{9} & +  0.03 x_{6}\\
 x_{10}   &  15.2295081967 & -4.97 x_{1} & -3.89 x_{2} & -7.15 x_{3} & -1.39 x_{9} & +  0.25 x_{6}\\
\hline
z    &  0.0  &   & -1.00 x_{2} & -1.00 x_{3} &    &   \\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  0.573770491803 & +  0.08 x_{1} & +  0.79 x_{2} & -0.87 x_{3} & +  0.02 x_{9} & +  0.11 x_{6}\\
 x_{7}   &  17.9016393443 & -6.44 x_{1} & -6.05 x_{2} & -0.51 x_{3} & +  0.31 x_{9} & -0.82 x_{6}\\
 x_{8}   &  17.6229508197 & -8.20 x_{1} & -12.69 x_{2} & -1.11 x_{3} & -0.64 x_{9} & -0.48 x_{6}\\
 x_{4}   &  0.16393442623 & -0.26 x_{1} & +  0.08 x_{2} & +  0.18 x_{3} & +  0.15 x_{9} & +  0.03 x_{6}\\
 x_{10}   &  15.2295081967 & -4.97 x_{1} & -3.89 x_{2} & -7.15 x_{3} & -1.39 x_{9} & +  0.25 x_{6}\\
\hline
z    &  0.0  &   & -1.00 x_{2} & -1.00 x_{3} &    &   \\
\end{array}\]
Final Dictionary
Solution:  0.0
Num Pivots:  1
\end{document}
